KVPY SX SB Exam 2018 Question Paper

Question : 84 of 120

Marks: +1, -0

The number of solutions x of the equation

\sin(x + x^2) – \sin(x^2) = \sin x

in the interval [2, 3] is

Solution:

\sin(x + x^2 )= \sin x+\sin(x^2 )

2\sin({x + x^2} /2) \cos({x + x^2} /2)

=2\sin({x + x^2} /2) \cos({x - x^2} /2)

2\sin({x + x^2} /2)(\cos({x + x^2} /2)

-\cos({x - x^2} /2))=0

-2\sin({x + x^2} /2).2\sin(x/2)\sin(x^2/2)=0

⇒

\sin x/2=0

\sin x^2/2=0

\sin({x + x^2} /2)=0

⇒

x = 2π

x =√{2π}

{x + x^2} /2 =π

x^ 2 +x -2 π = 0

x={√{1+8π}-1}/2∈ (2,3)

and

x={√{1+8π}+1}>3

x =√{ 2π}

and

x={√{1+8π}-1}/2∈ (2,3)

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